Defining the Similarity of Dissolution Profiles Using Hotelling's T2 Statistic

In today's drug development industry, establishing the similarity of dissolution profiles is a regulatory requirement. To derive meaningful conclusions, current practice evaluates the entire dissolution profile, compared with the earlier approach that used one time point (for example, the time required for 90% drug dissolution for both reference and test product).

The US Food and Drug Administration's (FDA's) guidance for industry on dissolution testing of immediate-release, solid, oral dosage forms1 describes the model-independent mathematical approach proposed by Moore and Flanner2 for calculating a dissimilarity factor (f1) and a similarity factor (f2) of dissolution across a suitable time interval. The similarity factor f2 (where 0<f2<100 and f2>50%, dissolution profiles are defined as similar) is a function of the mean differences and does not take into account the differences in dissolution within the test and reference batches. Hence, careful interpretation is warranted when f2 is used as a similarity factor with a large difference in variances between the two profiles.

Previous work
Previous articles have discussed the more serious deficiencies of using the f2 factor for assessing the similarity between two profiles. One of the major drawbacks identified was finding the sampling distribution of the statistic. This statistic has complicated properties, and deriving the distribution of the statistic is not mathematically tractable. Shah et al. proposed a bootstrap method to simulate a confidence interval for the f2 factor.3 Because the f2 is sensitive to the measurements obtained after either the test or reference batch has dissolved more than 85%, Shah and co-workers recommended a limit of one sampling time point after 85% dissolution.

Several other authors have discussed defining criteria for dissolution similarity using model-dependent as well as model-independent approaches. For example, Tsong proposed modelling profiles for individual tablets, and establishing specifications for similarity based on joint confidence regions for level and shape parameters of both reference and test batches.4 The disadvantage of this approach is that such regions are hard to interpret and shrink as a function of the amount of reference material tested. Sathe et al. also discussed a model-dependent approach using Mahalanobis distance,5 and Chow and colleagues proposed dissolution difference measurement and similarity testing based on a time series model in this context.6 However, dissolution data collected for profile comparisons typically have a very limited number of unequally spaced sampling intervals, whereas the intended time series has a large number of equally spaced sampling intervals.

Previously, Gohel and co-workers proposed another model-independent method based on the average absolute value of the log of the ratio of the area under the dissolution curve of both test and reference drugs.7

In the FDA guideline for industry, the procedure (f2 factor) allows the use of mean data and recommends that the per cent coefficient of variation at an earlier time point (for example, 15 min) not be more than 20%, and at other time points not more than 10%. In instances in which the per cent coefficient of variation within a batch is more than 15%, the industry guidelines suggest using a multivariate model-independent procedure.

This article examines the use of a multivariate procedure for testing the equivalence of two dissolution profiles through Hotelling's T2 statistic and compares this procedure with the f2 closeness criterion for different variance-covariance structures through simulation studies for normal and non-normal distributions. The two methods were also compared using real data examples.

Multivariate approach
Suppose xij;Np(mj,S), j 5 1,2 and i = 1,2 . . . n are two independent multivariate normal dissolution profiles, one being a reference batch or a prechange batch and the other being a test batch or a postchange batch. Samples were taken at p different time points; n is the number of vessels, either 6 or 12, which is typical for tablet dissolution for a pharmaceutical product. The dissolution time points (p) for both profiles should be the same. Let ¯x1, ¯x2 be the vector of the sample mean for each of the profiles; S is the sample variance -- covariance matrix, and n1, n2 are the number of tablets tested from each batch. To define a closeness of the two dissolution profiles, (12a) 100% confidence region for m1-m2 is considered, assuming the data follow a multivariate normal distribution. The confidence region for m1-m2 (5m) with confidence level 12a is the set of vectors m satisfying

where Fp,n2p11 (a) is the (12a) 100% percentile of the central F distribution with p and n2p11 degrees of freedom and n(5 n11n222) is the within-sample degrees of freedom. Assume m 5 dJp, where Jp is a p dimensional column vector of 1's. The maximum mean difference (d 5 max d) for which the equivalent of two given profiles could be concluded with (12a) 100% confidence given the data are the solution of d to equation 2.